Semistability and Restrictions of tangent bundle to curves

We consider all complex projective manifolds X that satisfy at least one of the following three conditions:

1. (1)
   There exists a pair \({(C\,,\varphi)}\) , where C is a compact connected Riemann surface and

   $\varphi\,:\, C\,\longrightarrow\, X$

   a holomorphic map, such that the pull back \({\varphi^* {\it TX}}\) is not semistable.
    
2. (2)
   The variety X admits an étale covering by an abelian variety.
    
3. (3)
   The dimension dim X ≤ 1.
    

We prove that the following classes are among those that are of the above type.

• All X with a finite fundamental group.
• All X such that there is a nonconstant morphism from \({{\mathbb C}{\mathbb P}^1}\) to X.
• All X such that the canonical line bundle K _X is either positive or negative or \({c_1(K_X)\,\in\,H^2(X,\, {\mathbb Q})}\) vanishes.
• All X with \({{\rm dim}_{\mathbb C} X\, =\,2}\).